9/23/2002, VECTORS O. Knill, Math 21aVECTORS. Two points P1= (x1, y1, z1), Q = P2= (x2, y2, z2) determine a vector v = (x2−x1, y2−y1, z2−z1).It points from P1to P2and we can write P1+ v = P2.Points P in space are in one to one correspondence to vectors pointing from 0 to P . The numbers viin a vectorv = (v1, v2, v3) are also called coordinates of the vector.REMARK: vectors can be drawn everywhere in space. If a vector starts at 0, then the vector v = (v1, v2, v3)points to the point (v1, v2, v3). That’s is why one can identify points P = (a, b, c) in space with a vectorv = (a, b, c). Two vectors which are translates of each other are considered equal. (∗1)1ADDITION SUBTRACTION, SCALAR MULTIPLICATION.xyuvu+vxyuvu-vxyu3 uBASIS VECTORS. The vectors i = (1, 0, 0), j = (0, 1, 0) and k = (0, 0, 1) are called basis vectors.Every vector v = (v1, v2, v3) can be written as a sum of basis vectors: v = v1i + v2j + v3k.WHERE DO VECTORS OCCUR?Velocity: if (f(t), g(t), h(t)) is a curve, then v =(f0(t), g0(t), h0(t)) is the velocity vector at the point(f(t), g(t), h(t)).Forces: static problems involve the determination of a force on objects.Fields: fields like electromagneticor gravitational fields or velocityfields in fluids are described withvectors.1The remark is a common point of confusion. Mathematicians call vectors affine vectors and restrict the word vectors to affinevectors attached to zero. Calculus courses don’t want to add too much terminology and call affine vectors simply vectors. Sometimes,vectors attached to 0 are called bound vectors. Most courses (like this one) opt for more simplicity and use only the word vectors -considering the resulting confusion not grave enough to worry about.Qbits: in quantum computation,one does not work with bits, butwith qbits, which are vectors.Color Any color can be written asa vector v = (r, g, b), where r is thered component, g is the green com-ponent and b is the blue component.redgreenblue(r,g,b)SVG Scalable Vector Graphics is an emerging standard for theweb. From www.w3.org: ”SVG is a language for describingtwo-dimensional graphics in XML. SVG allows for three typesof graphic objects: vector graphic shapes (e.g., paths consistingof straight lines and curves), images and text. Graphical ob-jects can be grouped, styled, transformed and composited intopreviously rendered objects. The feature set includes nestedtransformations, clipping paths, alpha masks, filter effects andtemplate objects.VECTOR OPERATIONS: The ad-dition and scalar multiplication ofvectors satisfies some properties.They are all ”obvious” (there is nopoint in memorizing them).u + v = v + u commutativityu + (v + w) = (u + v) + w additive associativityu + 0 = u + 0 null vectorr ∗ (s ∗ v) = (r ∗ s) ∗ v scalar associativity(r + s)v = v(r + s) distributivity in scalarr(v + w) = rv + rw distributivity in vector1 ∗ v = v one elementLENGTH OF A VECTOR.The length ||v|| of a vector v is the distance from the beginning to the end of the vector.EXAMPLE. The length of the vector v = (3, 4, 5) is ||v|| =√50 = 5√2.TRIANGLE INEQUALITY: ||u + v|| ≤ ||u|| + ||v||.UNIT VECTOR.A vector of length 1 is called a unit vector. If v is a vector which is not zero, then v/||v|| is a unit vector.EXAMPLE: If v = (3, 4), then v = (2/5, 3/5) is a unit vector in the plane.PARALLEL VECTORS.Two vectors v and w are called parallel, if v = rw with some constant
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